Single Equations Can Draw Pictures

Abstract

Analytic geometry is more powerful than Euclidean geometry, but students find that analytic geometry leaves many of the familiar objects of Euclidean geometry behind. Thus, any ellipse has a polynomial defining it, but where's the polynomial defining a triangle or a square or two intersecting circles? The purpose of this paper is to bring the two worlds of Euclidean and analytic geometry a little closer by one of the themes of higher mathematics?the versatility of polynomials. In high school I'd just learned the equation of a circle. During a school assembly a friend and I tried unsuccessfully to find a single equation that would give two concentric circles. The next day my friend appeared in math class like one bearing a rare treasure. He'd asked his father who, conveniently, was a mathematics professor. On a piece of paper there was neatly written: The equation of two circles of radii 1 and 2, centered at the origin, is